Minimal coadjoint orbits and symplectic induction

Abstract

Let $(X,\omega )$ be an integral symplectic manifold and let $(L,\nabla )$ be a quantum line bundle, with connection, over $X$ having $\omega$ as curvature. With this data one can define an induced symplectic manifold $(\tilde{X},{\omega }_{\tilde{X}})$ where $dim\tilde{X}=2+dimX$. It is then shown that prequantization on $X$ becomes classical Poisson bracket on $\tilde{X}$. We consider the possibility that if $X$ is the coadjoint orbit of a Lie group $K$ then $\tilde{X}$ is the coadjoint orbit of some larger Lie group $G$. We show that this is the case if $G$ is a non-compact simple Lie group with a finite center and $K$ is the maximal compact subgroup of $G$. The coadjoint orbit $X$ arises (Borel-Weil) from the action of $K$ on $\p$ where $\g= \k +\p$ is a Cartan decomposition. Using the Kostant-Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic isomorphism $(\tilde{X},{\omega }_{\tilde{X}})\cong (Z,{\omega }_Z)$ where $Z$ is a non-zero minimal "nilpotent" coadjoint orbit of $G$. This is applied to show that the split forms of the 5 exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.

38 pages, plain tex